Because of its internal battery, the NTG2 is going to stick out pretty far or, depending how you mount it stick back and interfere with the handgrip/ handle.

I use an NTG1 which is shorter along with a right angle coiled cable and it's acceptable. However, with a deadcat, it gets into frame at a certain point at the wide end. I wish I had a shorter mic. The only time I miss the internal battery is when it would be convenient to stick a plug-on for a wireless shotgun. But day to day, the NTG2 length would be aggravating to me. YMMV

These length issues can be addressed with some of the alternative mounting options that move the mic further off the camera. I bought my NTG1 years ago. I think there are some alternative mics now that are good and shorter.

I use an NTG-2 and a NTG-3 (which is only about an inch shorter). To move the microphone back, I use a Rode SM5 mount turning out the back of the camera's mount which move the front of the mic even farther back so there are no problems with it sticking out too far even with the deadcat. I wrap a rubber band around the SM5 to make it fit the EX1r's mount -- otherwise it's a little small.

If you'd like a picture of the setup let me know, and I'll post one here for you.

The NTG-2 is a great mic and will probably work fine for you, but the NTG-3 is substantially better and it was worth the money to get noticeably improved sound. (I don't have the best ear for sound, but it is still very noticeable to me.) I rarely use my NTG-2 anymore.

I also liked my Sennheiser ME-66 a lot. I sold it because I just didn't need another mic with the Rode mics covering both ends of the spectrum.

If you shoot indoors, a Sennheiser ME-64 is a very good choice, too. It uses the same power module that screws on as the ME66.

There are lots of comparison videos on Vimeo and YouTube for these mics. They were very helpful to me when researching purchases.

... I wrap a rubber band around the SM5 to make it fit the EX1r's mount -- otherwise it's a little small....

A #14 o-ring from the plumbing department at your favorite hardware store works perfectly for the NTG-1 in the EX1r mount. I don't know about the SM5 but o-rings make for a clean and even grip along the whole length using as many o-rings as you need.

I absolutely hate shotgun mics on camera...they're really not THAT good at pulling sound from a distance and for close ambient they're too directional.
My opinion is that they belong on a boom in outdoor situations, not on camera. It's a specific tool which many people use...well, wrong!

Rode NT5 cardioid is on my EX1 and I'm very satisfied plus it's a high spl mic and it will take some serious sound levels to clip it. No matter how loud the ambient is it will sound clear and without distortion. I know for a fact that it will sound MUCH cleaner than NTG1 in loud ambient, although the specs say there's only 4 dB difference in max SPL (143 vs 139).

Although there's a drawback, it will pick up low frequency handling noise of the camera so when you're handholding in a really quiet ambient you need to engage the low cut. Or find a way to mount a shockmount on the camera. I find it too cumbersome.

I've been using the Sennheiser MKH 416 on my EX1R, and I much prefer it to the camera's built-in mic. For wind/noise control I've got a Rycote softie on the microphone, and am using J-Rod's twin mic-mount on the camera (in place of the plastic mic mount that comes with the camera), and K-Tek long shotgun rubber isolation mount. So far it's worked out quite well, and I much prefer the audio from the MKH-416 as compared to the audio from the built-in microphone.

One very nice feature of the EX1R is the ability to 'gang' the audio internally (via a menu setup) such that you can record the monaural MKH 416 to both camera audio tracks, and set each track's input with different padding/level. Later when working with the single channel of audio I can pick up quieter sounds from the higher-gain channel, and switch back to the lower gain track for loud sounds.

I've also experimented with running the 416 and 816 microphones into my Sound Devices 702 recorder, and the audio is noticeably quieter on the Sound Devices recorder as compared to the EX1R. So I try to shoot double-system when possible (e.g., when an extra pair of hands is available). Synchronizing the wild audio tracks has not been a problem so far.

On top of the camera is NOT a good spot for ANY microphone. For 99% of applications is is too far away from the sound source. The only semi-legitimate applications for a camera-mounted microphone are literally "run-n-gun" newsgathering or maybe collecting impromptu comments at a casual wedding reception. I can't think of any other legitimate applications unless sound is really at the bottom of the list of importance to you.

...For 99% of applications is is too far away from the sound source. The only semi-legitimate applications for a camera-mounted microphone are literally "run-n-gun" ...

Certainly, mics that are too far away are undesirable. However, the usefulness of camera mounted shotguns in run and gun and the other situations you mentioned are not 1% of the time for some of us, maybe even most of us. It all depends on what you shoot or want to be able to handle in the future. As an example, I do interviews in a small studio where the camera is 3-4 feet off the subject and the camera shotgun as a backup to the boom shotgun has saved my bacon a couple times but it always provides some nice ambient sound. Still other times, when using wireless lavs and a field mixer tromping through the woods, it was there at the flick of a switch to catch impromptu sound, BTS and other stuff. YMMV

I have the AT875 and can say you will not be disappointed with it. The stock mic holder is prone to breaking at the least opportune moment. Replace it with the complete setup I have pointed out here will give you more flexibility, get the mic further away from camera noise, and is much more durable.

The J-Rod twin mount is the one that I put on my EX1R, it's been fine for me.

There are a pair of plastic registration pins beneath the plastic Sony mic mount, and the J-Rod aluminum mount has a slot machined in it to accomadate and align with the pins; I used a file to widen the mount's slot slightly, as there was interference with the pins (and I did not want to shave off the plastic pins).

Makes me think the J-Rod twin mount was designed for the EX1, and not updated to match changes in the EX1R (though I have not seen what's beneath an EX1 mic mount so I do not know for certain if there was a change by Sony).